All Questions
10 questions
11
votes
1
answer
393
views
Growing a chain of unit-area triangles: Fills the plane?
Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...
- 151k
4
votes
1
answer
193
views
A bound on the square distance of a random walk on undirected graph
Fact:
Let $G$ be an $n$-vertex undirected graph and $(X_s)_{s\in \mathbb N}$ a stationary random walk on $G$. Then for every $s\in \mathbb{N}$,
$ \mathbb{E}[d_G(X_s,X_0)^2] \le C s \log n $, for some ...
- 401
2
votes
1
answer
185
views
$\zeta(2n)$ and Levy processes
I am missing some steps in the final derivation of a probabilistic computation of the even values of $\zeta$. They show the Cauchy distribution is relate to a certian Levy process:
$$ |\mathbb{C}_1| \...
- 22.8k
3
votes
1
answer
202
views
Diameter of $n$-unit-vector closed scribble
Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...
- 151k
9
votes
1
answer
784
views
Width of a random convex polygon
Consider a planar (2D) random walk comprised of N steps.
Consider the minimum convex polygon enclosing the N points visited by the random walker.
Assume the definition of the width of a convex polygon ...
- 91
9
votes
1
answer
642
views
Twisted random walks
Suppose the points of two random walks in $\mathbb{R}^2$ are given the
step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$.
Here are several pairs of walks of $n=...
- 151k
1
vote
1
answer
555
views
The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
- 55
45
votes
1
answer
4k
views
Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...
- 151k
17
votes
4
answers
823
views
Sweep-segment bot: Will this random walk sweep the plane?
This model is inspired by the random behavior of the
Roomba sweeping robot.
Let a unit segment $ab$ in the plane be placed
initially with $a=(0,0)$ and $b=(1,0)$.
The segment is first rotated a ...
- 151k
16
votes
3
answers
2k
views
A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k